Cyclic Group Challenge

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Test your knowledge of cyclic groups and their properties! Answer whether each statement about cyclic groups is true or false. Can you distinguish the facts from the misconceptions?

Created by

Anna Arsolon

United States

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#	Player	Accuracy	time
1	Frias, Samantha Abigail B. April 28, 2025	% 30	00:36

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Cyclic Group Challenge


Cyclic Group ChallengeOnline version Test your knowledge of cyclic groups and their properties! Answer whether each statement about cyclic groups is true or false. Can you distinguish the facts from the misconceptions? by Anna Arsolon 1 A cyclic group is always abelian. Yes No 2 A cyclic group can only have one generator. Yes No 3 The group Z_8 under addition is a finite cyclic group. Yes No 4 The element 0 is always a generator of Z_(n). Yes No 5 Subgroups of a cyclic group are always cyclic. Yes No 6 Every finite cyclic group has a generator. Yes No 7 The generators of a cyclic group are always unique. Yes No 8 Every cyclic group is finite. Yes No 9 The group Z under addition is an infinite cyclic group. Yes No 10 Every element in a group generates the whole group. Yes No 11 A cyclic group cannot have subgroups. Yes No 12 The number of generators of a cyclic group of order n is given by Euler’s totient function ϕ(n). Yes No 13 The group Z under multiplication is a cyclic group. Yes No 14 A cyclic group can have multiple generators. Yes No 15 The order of a generator in a cyclic group is equal to the order of the group. Yes No 16 The identity element cannot be a generator of a cyclic group. Yes No 17 Cyclic groups can have non-abelian properties. Yes No 18 All abelian groups are cyclic. Yes No 19 Infinite cyclic groups do not cycle back or repeat elements. Yes No 20 Cyclic groups cannot be infinite. Yes No

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Cyclic Group Challenge

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About this activity

Created by

Download the paper version to play

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Cyclic Group ChallengeOnline version

by Anna Arsolon

A cyclic group is always abelian.

A cyclic group can only have one generator.

The group Z_8 under addition is a finite cyclic group.

The element 0 is always a generator of Z_(n).

Subgroups of a cyclic group are always cyclic.

Every finite cyclic group has a generator.

The generators of a cyclic group are always unique.

Every cyclic group is finite.

The group Z under addition is an infinite cyclic group.

Every element in a group generates the whole group.

A cyclic group cannot have subgroups.

The number of generators of a cyclic group of order n is given by Euler’s totient function ϕ(n).

The group Z under multiplication is a cyclic group.

A cyclic group can have multiple generators.

The order of a generator in a cyclic group is equal to the order of the group.

The identity element cannot be a generator of a cyclic group.

Cyclic groups can have non-abelian properties.

All abelian groups are cyclic.

Infinite cyclic groups do not cycle back or repeat elements.

Cyclic groups cannot be infinite.